Electromagnetic fields and anomalous transports in heavy-ion collisions— A pedagogical review

# Electromagnetic fields and anomalous transports in heavy-ion collisions — A pedagogical review

Xu-Guang Huang Physics Department and Center for Particle Physics and Field Theory, Fudan University, Shanghai 200433, China.
July 13, 2019
###### Abstract

The hot and dense matter generated in heavy-ion collisions may contain domains which are not invariant under P and CP transformations. Moreover, heavy-ion collisions can generate extremely strong magnetic fields as well as electric fields. The interplay between the electromagnetic field and triangle anomaly leads to a number of macroscopic quantum phenomena in these P- and CP-odd domains known as the anomalous transports. The purpose of the article is to give a pedagogical review of various properties of the electromagnetic fields, the anomalous transports phenomena, and their experimental signatures in heavy-ion collisions.

## I Introduction

As is well known, the strong interaction provides the mechanism that binds the quarks and gluons together to form the hadrons such as the proton and neutron. Our contemporary understanding of strong interaction is described by quantum chromodynamics (QCD) — a quantum gauge field theory based on color gauge symmetry. Despite its simple form, QCD possesses a number of remarkable properties among which the most mysterious one may be the color confinement: in vacuum, the quarks and gluons, as being colorful particles, are always confined in colorless hadrons. The color confinement property forbids us to observe isolated quark or gluon. However, at high temperature and/or high quark chemical potentials, the normal hadronic matter is expected to transform to deconfined quark-gluon matter. When the temperature is high such quark-gluon matter is usually called the quark-gluon plasma (QGP). Lattice QCD simulations which are the first-principle computations that solve QCD directly show that the “transition temperature” from hadronic matter to QGP is about MeV (at zero quark chemical potentials). It is believed that such a high temperature was once realized in the universe at a few microsecond after the Big Bang and the QGP was created at that time. In laboratory, up to now, the only method to achieve such a high temperature is to use the high-energy heavy-ion collisions. Such collisions have been carried out in the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and in the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN). The top center-of-mass energy per nucleon pair at RHIC Au + Au collisions is GeV and at LHC Pb + Pb collisions is TeV and will be upgraded to TeV soon. In these colliders, two nuclei are accelerated to velocity very close to the speed of light and then collide, the energy deposition in the reaction region can be large enough to create the hot and dense environment in which the deconfinement condition is reached. Measurements performed at RHIC and LHC have collected many signals supporting the generation of the QGP and have also revealed a variety of unusual properties of the QGP, e.g., its very small shear viscosity comparing to the entropy density and its high opacity for energetic jets.

Heavy-ion collisions can generate electromagnetic (EM) fields as well Rafelski:1975rf (). Recent numerical simulations found that the magnitude of the magnetic field in RHIC Au + Au collisions at GeV can be at the order of Gauss 111In the relevant literature, people usually use or MeV as the unit of where is the electron charge magnitude and MeV is the pion mass. In converting to the SI or Gaussian units, it is helpful to note the following relation: MeV Gauss (if , otherwise the right-hand side should be multiplied by ). and in LHC Pb + Pb collisions at TeV can reach the order of Gauss Kharzeev:2007jp (); Skokov:2009qp (); Voronyuk:2011jd (); Bzdak:2011yy (); Ou:2011fm (); Deng:2012pc (); Bloczynski:2012en (); Bloczynski:2013mca (); Zhong:2014cda (); Zhong:2014sua (). The electric filed can also be generated owing to event-by-event fluctuations Bzdak:2011yy (); Deng:2012pc (); Bloczynski:2012en (); Bloczynski:2013mca () (we will explain the physical meaning of such fluctuations in Sec. II) or in asymmetric collisions like Cu + Au collision Hirono:2012rt (); Deng:2014uja (); Voronyuk:2014rna (), and its strength is roughly of the same order as the magnetic field. Thus heavy-ion collisions provide a unique terrestrial environment to study QCD matter in strong EM fields. In particular, recently it was proposed that the magnetic field can convert topological fluctuations in the QCD vacuum into global electric charge separation along the direction of the magnetic field. The underlying mechanism is the so-called chiral magnetic effect (CME) Kharzeev:2007jp (); Fukushima:2008xe (). Some relatives of the CME were also proposed, including the chiral separation effect (CSE) Son:2004tq (); Metlitski:2005pr (), chiral electric separation effect (CESE) Huang:2013iia (), chiral magnetic waves (CMW) Kharzeev:2010gd (), chiral vortical effect (CVE) Erdmenger:2008rm (); Banerjee:2008th (); Son:2009tf (), chiral vortical wave Jiang:2015cva (), chiral heat wave Chernodub:2015gxa (), chiral Alfven wave Yamamoto:2015ria (), etc. They all represent special transport phenomena that are closely related to chiral anomaly 222We will not distinguish the terms “chiral anomaly”, “axial anomaly”, and “triangle anomaly” in this article. and thus are called anomalous transports. The experimental searches of the anomalous transports have been carried out at RHIC and LHC and the measurements indeed offered signals consistent with the predictions of the CME, CMW, and CVE; see the discussions in Sec. IV.

The purpose of this paper is to give a pedagogical review of recent progresses on the study of EM fields and the anomalous transport phenomena induced by EM fields in heavy-ion collisions. We will keep all the discussions as intuitive as possible and lead the readers who wish to understand more technical details to proper literature. To access this paper, the readers do not need to have expert knowledge of QCD; only elementary knowledge of quantum field theory and heavy-ion collisions are needed. (Perhaps the only exception is Sec. III.1 (3) where some knowledge of topology and gauge field theory is needed; we thus give more thorough discussion and put necessary references there so that the readers can easily trace the relevant literature.) Thus this paper will be particularly useful for graduate students who have finished their first-year courses and wish to enter the exciting research area of anomalous transport phenomena. In this aspect, the present review is complementary to existing excellent reviews, e.g., Refs. Tuchin:2013ie (); Kharzeev:2009fn (); Kharzeev:2013jha (); Kharzeev:2013ffa (); Kharzeev:2015kna (); Kharzeev:2015znc () in which more advanced materials can be found.

We organize the paper as follows. In Sec. II, we shall discuss some general properties of the EM fields in heavy-ion collisions. In Sec. III, we shall give an elementary introduction to the anomalous transports in parity-odd (P-odd) and/or charge-conjugation-odd (C-odd) medium. The experimental implications of the anomalous transports and the current status of their detection in heavy-ion collisions will be reviewed in Sec. IV. Some discussions will be presented in Sec. V.

In addition to the anomalous transports, the EM fields can drive a range of other intriguing phenomena including, for example, the magnetic catalysis of chiral symmetry breaking Gusynin:1994re (); Gusynin:1995nb (); Shovkovy:2012zn (), the inverse magnetic catalysis or magnetic inhibition at finite temperature and density Preis:2010cq (); Bali:2011qj (); Bali:2012zg (); Bruckmann:2013oba (); Fukushima:2012xw (); Fukushima:2012kc (); Kojo:2012js (); Chao:2013qpa (); Yu:2014sla (); Feng:2014bpa (); Yu:2014xoa (); Cao:2014uva (); Ferrer:2014qka (); Mueller:2015fka (); Chen:2015hfc (), the possible meson condensation in strong magnetic field Chernodub:2010qx (); Chernodub:2011mc (); Hidaka:2012mz (); Liu:2014uwa (); Liu:2015pna (), the neutral pion condensation in vaccum Cao:2015cka (), the anisotropic viscosities in hydrodynamic equations Braginskii:1965 (); Lifshitz:1981 (); Huang:2009ue (); Huang:2011dc (); Tuchin:2011jw (), and the early-stage phenomena in heavy-ion collisions like the EM-field induced particle production Tuchin:2010vs (); Tuchin:2010gx (); Tuchin:2012mf (); Tuchin:2013ie (); Tuchin:2014nda (); Tuchin:2014pka (); Basar:2012bp () and the dissociation of heavy-flavor mesons Marasinghe:2011bt (); Machado:2013rta (); Alford:2013jva (); Liu:2014ixa (); Guo:2015nsa (). These topics will not be the main focus of this article. Some of them are nicely reviewed in Refs. Kharzeev:2013jha (); Andersen:2014xxa (); Miransky:2015ava ().

## Ii Properties of electromagnetic fields in heavy-ion collisions

The reason why heavy-ion collisions can generate magnetic fields is simple: nuclei are positively charged and when they move they generate electric currents which in turn induce the magnetic fields. In a noncentral heavy-ion collision, two counter-propagating nuclei collide at a finite impact parameter ; one can easily imagine that the magnetic field at the center of the overlapping region will be perpendicular to the reaction plane owing to the left-right symmetry of the collision geometry (see Fig. 1 for illustration). However, in a real collision event, this left-right symmetry may be lost because the nucleon distribution of one nucleus would not be identical to another. We will come to this point later, but first let us estimate how strong the magnetic field can be.

Let us consider Au + Au collisions at fixed impact parameter fm and at RHIC energy GeV as an example. If we approximate the problem by assuming that all the protons are located at the center of the nucleus, then by naively applying the Biot-Savart law we obtain

 −eBy ∼ 2ZAuγe24πvz(2b)2≈10m2π≈1019Gauss, (1)

where ( is the nucleon mass) is the velocity of the nucleus, is the Lorentz gamma factor, and is the charge number of gold nucleus. The minus sign on the left-hand side is because the magnetic field in pointing to the direction in the setup shown in Fig. 1.

This is really a huge magnetic field. It is much larger than the masses squared of electron, , and light quarks, , and thus is capable of inducing significant quantum effects. It is also larger than the magnetic fields of neutron stars including the magnetars which may have surface magnetic fields of the order of Gauss Olausen:2013bpa (); Turolla:2015mwa (). Therefore the magnetic fields generated in high-energy heavy-ion collisions are among the strongest ones that we have ever known in current universe. (In the early universe, there was a possibility to generate an even stronger magnetic field through the electro-weak transition, see Ref. Grasso:2000wj () for review.) One can expect that such a huge magnetic field may have important consequences on the dynamics of the quark-gluon matter produced in heavy-ion collisions. We will discuss several such consequences, namely, the anomalous transport phenomena in Sec. III. In this section we will focus on the fields themselves.

### ii.1 Computations of the electromagnetic fields

The estimation given in Eq. (1) is too simplified, in order to have a more reliable simulation for the electromagnetic (EM) fields in heavy-ion collisions, the following issues need to be taken into account. (1) We need the knowledge of the proton and neutron distributions in a given nucleus. For this purpose, we can choose the well established Woods-Saxon distribution to use. (2) In real heavy-ion collisions, because the proton distribution varies from one nucleus to another, the generated EM fields vary from event to event. It is thus important to study the event-by-event fluctuation of the EM fields Bzdak:2011yy (); Deng:2012pc (); Bloczynski:2012en (); Bloczynski:2013mca (). (3) We need to replace the Biot-Savart law by the full relativistic Liénard-Wiechert potentials which contain the retardation effect,

 eE(t,r) = e24π∑nZnRn−Rnvn(Rn−Rn⋅vn)3(1−v2n), (2) eB(t,r) = e24π∑nZnvn×Rn(Rn−Rn⋅vn)3(1−v2n), (3)

where the summation is over all the charged particles, is the charge number of the th particle, is the relative position of the field point to the source point of the th particle, is the velocity of th particle at the retarded time . Note that Eqs. (2)-(3) have singularities at ; in practical calculations a variety of regularization schemes have been used and consistent results are obtained after taking the event average Skokov:2009qp (); Voronyuk:2011jd (); Bzdak:2011yy (); Deng:2012pc (); Bloczynski:2012en (); Bloczynski:2013mca (); Deng:2014uja ().

As the EM fields in heavy-ion collisions can be much larger than the electron and light quark masses squared, one may worry about the possible quantum electrodynamics (QED) correction to the otherwise classical Maxwell field equations (the Liénard-Wiechert potentials are the solutions of the Maxwell equations). So let us make a magnitude estimate of such QED correction by using the one-loop Euler-Heisenberg effective lagrangian for soft photons (see Ref. Dunne:2004nc () for review):

 LEH=−AμJμ−14FμνFμν−e232π2∫∞0dsse−sm2e[Recosh(esX)Imcosh(esX)Fμν~Fμν−4e2s2−23FμνFμν], (4)

where is the electric current, is the EM potential, is the strength tensor, , and . At strong-field limit, the asymptotic form of Eq. (4) behaves like Dunne:2004nc ()

 LEH∼−AμJμ−14[1−e224π2lne2|F2|m4e]FμνFμν, (5)

where . The field equations derived from this lagrangian can be regarded as the Maxwell equations but with a renormalized charge (keeping only the leading-log term because )

 e→~e≈e[1−e224π2lne2|F2|m4e]−1 (6)

at leading-log order. Thus, we can find that even for very strong EM field, e.g., , the quantum correction can only amend the final restuls by a few percent. This justifies the applicability of Eqs. (2)-(3).

In the following subsections we will review the recent results of the EM fields in heavy-ion collisions obtained by using the Liénard-Wiechert potentials on event-by-event basis. We will mainly focus on Au + Au collisions at RHIC and Pb + Pb collisions at LHC; other collision systems will be briefly discussed in Sec. II.8.

### ii.2 Impact parameter dependence

We first show in Fig. 2 the impact parameter dependence of the EM fields at and where the initial time is set to be the time when the two colliding nuclei completely overlap. The curves with full dots are for Au + Au collision at RHIC energy GeV and the curves with open dots are for fields scaled by a factor for Pb + Pb collision at LHC energy TeV. In these figures (and also in the figures hereafter) represents the average over events.

One can find that:
(1) The event averaged EM fields have only one nonzero component, ; all other components vanish: . However, owing to the fluctuations of the positions of protons in the nuclei, their magnitudes in each event can be large (except for the -components which are always small). This is reflected in the averaged absolute values of the fields and is most evident for central collisions Bzdak:2011yy (); Deng:2012pc ().
(2) When with the nucleus radius, the event-averaged field is proportional to and it reaches its maximum value around . The fluctuation-induced fields are not sensitive to when .

### ii.3 Collision energy dependence

As investigated in Ref. Bzdak:2011yy (); Deng:2012pc (), to high precision, the magnitudes of EM fields linearly depend on the collision energy . Actually the absolute values of the EM fields satisfy very well the following scaling law, where is a universal function which has the shapes for and as shown in Fig. 2. For the event-averaged magnetic field, , the following formula approximately expresses its impact parameter , collision energy , charge number and atomic number dependence:

 e⟨By⟩∝√s2mNZA2/3b2RAm2π,forb<2RA. (7)

Note that the prefactor is nothing but the Lorentz gamma factor.

### ii.4 Spatial distributions

The spatial distributions of the EM fields are evidently inhomogeneous. The contour plots for , , , and in the transverse plane at impact parameter fm and at for RHIC energy are shown in Fig. 3. The distributions of the fields for LHC energy is merely the same but with times larger magnitudes according to Sec. II.3.

One may notice that for noncentral collision, the -component of the electric field is very large along the -direction, reflecting the fact that at a large amount of net charges stays temporally in the center of the “almond”-shaped overlapping region. This strong, out-of-plane electric field may drive positive (negative) charges to move outward (toward) the reaction plane, and thus induce an electric quadrupole moment in the produced quark-gluon matter. Such an electric quadrupole moment, as argued in Ref. Burnier:2011bf (), may lead to an elliptic flow imbalance between and ; see Refs. Deng:2012pc (); Stephanov:2013tga () for the detail.

### ii.5 Azimuthal correlation with the participant planes

We have seen that the event-by-event fluctuations of the nuclear distribution can strongly modify the magnitudes of the EM fields, one then may ask how these event-by-event fluctuations affect the azimuthal orientations of the EM fields (see the illustrating Fig. 4). In fact, as revealed recently Bloczynski:2012en (), the event-by-event fluctuations generally make the magnetic fields unaligned with the normal direction of the reaction plane. As a consequence of the event-by-event fluctuations of nuclear distribution, the distribution of the participants (the nucleons that participate in the collision) in the overlapping region varies from event to event as well. Thus for each event, the overlapping region is not perfectly almond-shaped and its short-axis may be rendered away from the impact-parameter direction. Such shape and direction variations can be captured by the so-called eccentricity parameters and harmonic angles , . Mathematically, they are defined as

 ϵ1eiΨ1 = −∫d2rρ(r)r3eiϕ∫d2rρ(r)r3, (8) ϵneinΨn = −∫d2rρ(r)rneinϕ∫d2rρ(r)rn,n>1, (9)

where is the transverse distribution function of the participants. If there were no event-by-event fluctuations, for each event should be equal to . As we will see, the azimuthal orientation between of (and of ) and (the second harmonic angle of the participants) fluctuates with sizable spread in their relative angle about the expected value . (Note that this would imply that and with and the reaction plane angle and event plane angle also fluctuate.) This can be clearly seen from Fig. 5 which shows the histograms of over events for different . For the most central case the events are uniformly distributed indicating negligible correlation between and ; while for noncentral collisions the event distribution behaves like a Gaussian peaking at indicating correlation between and . For the following reason, such fluctuations in the correlation between (as well as ) and the participant planes may have important impacts on the experimentally measured quantities.

Let us consider the chiral magnetic effect (CME, see Sec. III.1) as a concrete example, but the analysis can be extended to the observables of other EM-field-induced transport phenomena. The CME contributes to the single particle distribution for charged hadrons a component with the charge which in turn contributes to the two-particle distribution the following term,

 fqαqβ∝qαqβ(eB)2cos(ϕα−ψB)cos(ϕβ−ψB), (10)

where are the azimuthal angles of the hadrons and . We therefore can extract the CME contribution to the two-particle correlation (which was used by STAR and ALICE collaborations to detect the charge separation with respect to the reaction plane, see Sec. IV.1) as

 γαβ∝qαqβ⟨(eB)2cos[2(ψB−Ψ2)]⟩. (11)

If the -direction were always perpendicular to the reaction plane while always coincide with (which we set to be zero here), then we simply have . But the fluctuations in magnetic field as well as in participant planes will blur the relative angle between the two and modify the signal by a factor . (Here we note that the magnitude of the magnetic field has no noticeable correlation to its azimuthal direction Bloczynski:2012en ().) Similarly, if one measures the two-particle correlation with respect to higher harmonic participant plane, for example, the fourth harmonic plane , , the azimuthal fluctuations of will again contribute a modification factor to it.

In Fig. 6 (left panel) one can find the computed average values of as functions of the impact parameter from the event-by-event determination of the -field direction at the collision center and the participants harmonics, , . The plots suggest:
(1) The correlations between and the odd harmonics are practically zero (as a consequence of parity invariance), while the correlations of with even harmonics are nonzero but suppressed comparing to the non-fluctuating case.
(2) The centrality dependence of agrees with the patterns shown in the histograms Fig. 5: it is significantly suppressed in the most central and most peripheral collisions indicating no correlations between and while is still sizable for moderate values of .
(3) As checked in Ref. Bloczynski:2012en (), there is no visible difference between the -weighted correlation and the unweighted correlation for . This indicates that the magnitude of the magnetic field does not noticeably correlate to its azimuthal direction.

The event-by-event fluctuations also bring modification to the correlations between -field orientation and the participant planes. In parallel to the -field case, in Fig. 6 (right panel), we show the correlations , , as functions of the impact parameter. It is seen that:
(1) There is a sizable negative correlation (i.e., back-to-back) between and which is strongest in the central collisions. This is simply because the pole of with more matter will concurrently have more positive charges from protons which induce the -field pointing opposite to .
(2) There is also a weak correlation between and .
(3) Similar to the -field case, -weighted correlations have no visible difference from the unweighted correlations indicating no correlation between -field magnitude and orientation.

### ii.6 Early-stage time evolution

In a high-energy heavy-ion collision, right after the collision, the produced partonic matter is mainly consist of gluons and is in a far-from-equilibrium state. This partonic matter subsequently evolves toward thermal equilibrium and a large number of quarks and anti-quarks are excited during this thermalization process. Although so far we still lack a theory to quantitatively understand the thermalization problem, the phenomenological studies revealed that the time scale of the completion of the thermalization is very short comparing to the total lifetime of the thermalized quark-gluon plasma (QGP). The relevant information can be found in the review articles, Refs. Arnold:2007pg (); Gelis:2012ri (); Huang:2014iwa (); Berges:2015kfa (). Once the thermalization is locally achieved, the bulk evolution of the system can be well described by hydrodynamics. One of the transport coefficients of the hydrodynamics, namely, the electric conductivity has been numerically simulated by using lattice QCD recently and it was found that for QGP is very large (see next subsection). A large makes the QGP sensitive to the variation of the EM fields and which in turn strongly influence the time evolution of the EM fields themselves. Thus the time evolution of the EM fields in the QGP stage need special treatments and we leave this issue to next subsection. In this subsection we will focus on the stage before the thermalization is achieved (we call this stage the “early stage”). The quark-gluon matter in the early stage is expected to be much less conducting than that in the QGP stage and we will simply assume it is insulating and thus ignore the response of the matter to the EM fields.

In Fig. 7 one can find the numerical results of the early-stage time evolutions of the EM fields at in collisions with fm for Au + Au collision at GeV and for Pb + Pb collision at TeV. The results are from Ref. Deng:2012pc (). The contributions to the EM fields come from the charged particles in spectators, participants, and remnants. We can see that the transverse fields decay very fast after the collision reflecting the fact that the spectators are leaving the collision region very fast. Once the spectators are all far away from the collision region, the remnants which moves much slower than the spectators become important and they essentially slow down the decays of the transverse fields. The lifetime of the magnetic fields due to spectators can be estimated as

 tB≈RA/(γvz)≈2mN√sRA, (12)

which is just half the time that one proton needs to pass through the nucleus freely. The lifetime is very short for large : for Au + Au collision at 200 GeV, fm, while for Pb + Pb collision at 2.76 TeV, fm. Within the time period the fields decay slowly and when , we can approximate the early-stage time evolution of the event-averaged magnetic field, that is, as

 ⟨eBy(t)⟩≈⟨eBy(0)⟩(1+t2/t2B)3/2. (13)

This formula works better for larger impact parameter and larger . As seen from Fig. 7, Eq. (13) fit the simulation results for very well for time ; after that the remnants dominate and the separation between the curves from Eq. (13) and from the simulations are visible. The Eq. (13) shows that for the magnetic field decays fast,  333One should note that the magnitude of the magnetic field is still very large even at ; for example, MeV for RHIC Au + Au collisions at fm which is still comparable to the light quark mass squared, .. However, if at that time the QGP has been already formed, its EM response will significantly modify the time evolution of the fields.

### ii.7 Late-stage (QGP-stage) time evolution

The discussions and simulations presented in the last subsection are based on the assumption that the produced matter is ideally insulting. This assumption is adoptable only in the early stage where the system is gluon-dominated but becomes less and less justified as the system evolves and more and more quarks and anti-quarks emerge. As a matter of fact, the QGP is a good conductor according to the theoretical and lattice QCD studies. At very high temperature the perturbative study gives that the electric conductivity of QGP is  Arnold:2003zc (). An old lattice calculation with found that  Gupta:2003zh () at with the deconfinement temperature. Another quenched lattice simulation using staggered fermions found that  Aarts:2007wj (). Recent quenched lattice studies using Wilson fermions obtained that  Ding:2010ga (); Francis:2011bt (); Ding:2014dua () for temperature . The lattice calculation with dynamical Wilson fermions found that at MeV Brandt:2012jc (). Another new lattice simulation using fermions obtained that for temperature  Amato:2013naa (); Aarts:2014nba (). In these results, the EM vertex parameter and is the charge of quark with flavor ; for example, if quarks are considered while if quarks are considered. Note that the deconfinement temperature is different in and cases; for example, if we have MeV while if we have MeV.

At MeV and choosing with quarks contributing to , one can find that the resulted is about times larger than that of copper at room temperature ( MeV at ).

Now let us analyze how the large influences the time evolution of the EM fields in the QGP stage which we refer to as “late stage”.

Our discussion will be based on magnetohydrodynamics. We first write down the Maxwell’s equations,

 ∇×E=−∂B∂t, (14) ∇×B=∂E∂t+J, (15)

where is the electric current. We treat the QGP as being locally charge neutral but conducting, thus is the sum of the external one and the one determined by the Ohm’s law,

 J=σ(E+v×B)+Jext, (16)

where is the flow velocity of QGP and is the current due to the motion of unwounded protons (most are in spectators). Using Eq. (16), we can rewrite the Maxwell’s equations as magnetohydrodynamic equations

 ∂B∂t=∇×(v×B)+1σ(∇2B−∂2B∂t2+∇×Jext), (17) ∂E∂t+∂v∂t×B=v×(∇×E)+1σ(∇2E−∂2E∂t2−Jext∂t), (18)

where we have used the Gauss laws and . Equation (17) is the induction equation, which plays a central role in describing the dynamo mechanism of stellar magnetic field generation. The first terms on the right-hand sides of Eqs. (17)-(18) are the convection terms, while the remained terms are called “diffusion terms” although they are not exactly in the diffusion-equation type. Let us discuss some outcomes of these magnetohydrodynamic equations.

(1) If , that is, if the QGP does not flow, the Eq. (17) reduces to

 ∂B∂t=1σ(∇2B−∂2B∂t2+∇×Jext). (19)

This equation can be solved by using the method of Green’s function, the details can be found in Refs. Tuchin:2013ie (); Tuchin:2013apa (); Tuchin:2014iua (); Tuchin:2014hza (); Gursoy:2014aka (); Zakharov:2014dia () in which the authors studied how the spectators induced magnetic field evolve in QGP phase (assuming the system is already in the QGP phase at the initial time). The main information from these studies are that the presence of the conducting matter can significantly delay the decay of the magnetic field. This is easily understood as the consequence of the Faraday induction: a fast decaying external magnetic field induces a circular electric current in the medium which in turn causes a magnetic field that compensates the decaying external magnetic field.

For late times, the external current from the spectators can be neglected (we will always assume this case in this subsection hereafter). If with the characteristic time scale over which the field strongly varies, one can neglect the second-order time derivative term and render Eq. (19) a diffusion equation:

 ∂B∂t=1σ∇2B. (20)

This case was studied in Ref. Tuchin:2010vs (). This equation describes the decay of the field due to diffusion, and the diffusion time of the magnetic field is given by

 tD=L2σ, (21)

with a characteristic length scale of the system over which the magnetic field varies strongly. Upon setting fm and MeV at MeV, the diffusion time is about fm. However, as argued by Mclerran and Skokov McLerran:2013hla (), in this case the condition is not satisfied, so it is more realistic to solve Eq. (19) instead its diffusion-type simplification.

(2) If and the magnetic Renolds number (the magnetic Renolds number quantifies the ratio of the convection term over the “diffusion term”), we can approximately keep only the convection terms in Eqs. (17)-(18). This corresponds to the ideally conducting limit. The equations such obtained are

 ∂B∂t=∇×(v×B), (22) E=−v×B. (23)

It is well-known that Eq. (22) leads to the frozen-in theorem for ideally conducting plasma, i.e., the magnetic lines are frozen in the plasma elements or more precisely the magnetic flux through a closed loop defined by plasma elements keeps constant. Thus the decay of the fields are totaly due to the expansion of the QGP. To see the consequence of Eqs. (22)-(23), we assume for simplicity a initial Gaussian transverse entropy density profile

 s(x,y)=s0exp(−x22a2x−y22a2y), (24)

where are the root-mean-square widths of the transverse distribution. They are of order of the nuclei radii if the impact parameter is not large. For example, for Au + Au collision at RHIC, fm for , and fm, fm for fm. By assuming the Bjorken longitudinal expansion,

 vz=zt, (25)

one can solve the ideal hydrodynamic equations for transverse expansion and obtain Ollitrault:2007du (),

 vx = c2sa2xxt, (26) vy = c2sa2yyt, (27)

where is the speed of sound. Substituting the velocity fields into Eq. (22), we can solve out analytically. For example, the at is give by

 By(t,0) = t0te−c2s2a2x(t2−t20)By(t0,0). (28)

This is just manifestation of the frozen-in theorem, because the areas of the cross section of the QGP expands according to in plane, thus the total flux across the plane is a constant. Setting fm and , we see from Eq. (28) that for fm decays approximately as — much slower than the -type decay in the insulating case discussed in last subsection.

So far, we discussed two special cases of Eqs. (17)-(18) which permit analytical treatments. It is desirable to solve the most general equations in accompanying with the hydrodynamic or kinetic-equation simulation for the, i.e. the fluid velocity and temperature, of the fireball. But up to today, this is not done yet.

### ii.8 Electromagnetic fields in other collision systems

In recent years, RHIC has also run heavy-ion collisions of nuclei other than gold, for example, the Cu + Au and U + U collisions. The EM fields in these collision systems were also studied Hirono:2012rt (); Bloczynski:2013mca (); Deng:2014uja (); Voronyuk:2014rna (); Chatterjee:2014sea (). We here give a brief summary of these studies.

The EM fields in U + U collisions were studied thoroughly in Ref. Bloczynski:2013mca (). The uranium U nucleus, unlike the Au or Pb nucleus, has a highly deformed prolate shape. But this shape deformation does not bring much effect to the event-averaged magnetic fields. As simulated in Ref. Bloczynski:2013mca (), the U + U collisions at GeV produce an event-averaged magnetic field just lightly smaller than that in Au + Au collisions at GeV, see the left panel of Fig. 8. The readers can find more information, especially those related to the event-by-event fluctuation, in Ref. Bloczynski:2013mca ().

The Cu + Au collisions are geometrically asymmetric: both the charge number and the total size of the gold nucleus are much larger than that of copper nucleus. Thus the Cu + Au collisions may be able to produce nonzero electric fields along the in-plane Au-to-Cu direction even after the event average. The numerical simulation presented in Ref. Deng:2014uja () found that:
(1) The strength of the event averaged magnetic field (which is along the direction) in Cu + Au collisions at 200 GeV is comparable to that in Au + Au collisions.
(2) There is a strong event averaged electric field pointing from the Au nucleus to Cu nucleus which is at the order of one , see the right panel of Fig. 8.
(3) The azimuthal angle of the electric field, , has a strong back-to-back correlation with , the first harmonic angle of the participants; this is the same as that in Au + Au collisions Bloczynski:2012en (). The new feature is that in noncentral Cu + Au collisions there is a clear positive correlation between and signaling a persistent in-plane electric field. More details can be found in Ref. Deng:2014uja ().

Very recently, there were interesting proposals for the novel effects of this in-plane field in Cu + Au collisions, for example: the field can lead to a directed flow splitting between positively and negatively charged hadrons Hirono:2012rt (); Voronyuk:2014rna (), the presence of the in-plane field may strongly suppress or even reverse the sign of the charge-dependent correlation (see Eq. (78) for its definition) Deng:2014uja (), and Cu + Au may serve to test the chiral electric separation effect Ma:2015isa (); more relevant discussions are given in Sec. IV.

## Iii Anomalous transports in P- and C-odd quark-gluon plasma

The strong EM fields may induce a variety of novel effects to the quark-gluon plasma, among which we will focus in this section on the ones that are deeply related to the topology and symmetry of QCD and QED. It was found that, in addition to the normal electric current driven by field, there can emerge three new currents in P- and C-odd regions in QGP as responses to the applied EM fields. They are the chiral magnetic effect (CME) Kharzeev:2007jp (); Fukushima:2008xe (), the chiral separation effect (CSE) Son:2004tq (); Metlitski:2005pr (), and the chiral electric separation effect (CESE) Huang:2013iia (). Thus the complete response of the P- and C-odd QGP to external EM field can be expressed as:

 (JVJA)=(σVVσVAσAVσAA)(EB), (29)

where and represent vector and axial currents and ’s are corresponding conductivities. The Ohm’s law and the conductivity  444We will specifically refer to the vector current as the electric current unless otherwise stated. Then is then just the usual electric conductivity which we denoted by in last section. is physically well understood, so we will not discuss them. In this section, we will focus on the CME, CSE, and CESE which are anomalous in the sense that their appearances are closely related to the topologically nontrivial vacuum structure of QCD and the axial anomaly. We will discuss their experimental consequences in next section.

### iii.1 Chiral magnetic effect

(1) What is the chiral magnetic effect? — The CME is the generation of vector current by external magnetic field in chirality-imbalanced (P-odd) medium. Historically, the CME has been studied through different theoretical approaches in a number of contexts ranging from astrophysics Vilenkin:1980fu (), condensed matter systems Nielsen:1983rb (); Alekseev:1998ds (); Volovik:2003fe (); Zyuzin:2012tv (); Kharzeev:2012dc (); Chen:2013mea (); Basar:2013iaa (); Landsteiner:2013sja (); Huang:2015mga (), QCD physics Kharzeev:2004ey (); Kharzeev:2007tn (); Kharzeev:2007jp (); Fukushima:2008xe (); Kharzeev:2009pj (); Fukushima:2009ft (); Asakawa:2010bu (); Fukushima:2010vw (); Fukushima:2010zza (); Brits:2010pw (); Hou:2011ze (); Kharzeev:2011ds () , to holographic models Newman:2005hd (); Yee:2009vw (); Rebhan:2009vc (); D'Hoker:2009bc (); Rubakov:2010qi (); Gorsky:2010xu (); Gynther:2010ed (); Hoyos:2011us (); Amado:2011zx (); Nair:2011mk (); Kalaydzhyan:2011vx (); Loganayagam:2012pz (); Lin:2013sga (). The recent reviews of CME are Kharzeev:2009fn (); Kharzeev:2013ffa (); Fukushima:2012vr (); Kharzeev:2015kna ().

The CME can be neatly expressed as

 JV = σVAB, (30) σVA = e22π2μA, (31)

for each specie of massless fermions with charge , where the current is defined by and is a parameter that characterizes the chirality imbalance of the medium. The is commonly called aixal or chiral chemical potential although it actually does not conjugate to any conserved charges of the fermions; we will discuss its meaning later. For QGP, the total CME current is obtained by adding up all the light quark’s contributions and the CME conductivity should be with the charge of quark of flavor and the number of color.

From Eq. (30), we can first recognize that CME is P odd because (a P-odd quantity) and (a P-even quantity) transform differently under parity. Thus CME can occur only in P-odd medium characterized by finite . Second, the CME is C even as and are both C-odd. Third, the CME is T-even as both and are T-odd 555Here, “T” stands for “time reversal”.; this is also evident from the fact that the CME conductivity , as expressed in Eq. (31), is temperature independent. The time-reversal-evenness of the CME conductivity indicates that the emergence the the CME current is a non-dissipative phenomenon Kharzeev:2011ds (); Kharzeev:2013ffa (). (Note that the usual electric conductivity is T-odd and thus can generate entropy.)

To get an intuitive understanding of Eq. (30), let us consider a system with unequal numbers of right-handed (RH) and left-handed (LH) quarks (for example, consider that with the total number of RHLH quarks) subject to uniform magnetic field. We know that the Landau quantization has the property that the lowest Landau level permits only one spin polarization which minimizes the excitation energy of quarks but is highly degenerate with a degeneracy factor proportional to the total magnetic flux. Thus if the magnetic field is strong enough so that this degeneracy factor is larger than or , all the quarks are confined to the lowest Landau level on which their spins are totally polarized to be along the direction of the magnetic field. Now the RH quarks will prefer to move along their spin direction that is the direction of the magnetic field; while the LH quarks will prefer to move opposite to their spin direction which is opposite to the direction of the magnetic field. Because the number of RH quarks are larger than the number of LH quarks, the overall effect of the motion of quarks will be to generate a net current of along the magnetic field. For quark, similar argument leads to a net current of moving opposite to the magnetic field or a net electric current along the direction of the magnetic field 666One should be noticed that the antipartile of a RH-chirality (RH-helicity) massless fermion is of LH-chirality (RH-helicity). This is the intuitive picture of the CME. Note that if the magnetic field is not strong enough so that the higher Landau levels are also occupied, then each higher Landau level will contain equal numbers of spin-up and spin-down quarks and the CME current of spin-up quarks will exactly cancel the CME current of spin-down quarks for each higher Landau level. Therefore the higher Landau levels do not contribute to the total current — only the lowest Landau level is responsible to the CME.

(2) How to derive the CME? — The emergence of the CME, Eq. (30), is due to the axial anomaly in QED sector which couples the vector current to the magnetic field and the axial chemical potential , see Fig. 9. There are a variety of methods to derive Eq. (30) from microscopic quantum field theory Vilenkin:1980fu (); Nielsen:1983rb (); Alekseev:1998ds (); Kharzeev:2007jp (); Fukushima:2008xe (); Fukushima:2009ft (); Fukushima:2010vw (); Brits:2010pw (); Hou:2011ze (); Zahed:2012yu (); Warringa:2012bq (), mesoscopic kinetic theory Son:2012wh (); Stephanov:2012ki (); Gao:2012ix (); Chen:2012ca (); Son:2012zy (); Chen:2013dca (); Dwivedi:2013dea (); Chen:2013iga (); Chen:2014cla (); Akamatsu:2014yza (); Manuel:2013zaa (); Manuel:2014dza (); Duval:2014ppa (); Chen:2015gta (); Gao:2015zka (), to macroscopic hydrodynamic approach Son:2009tf (); Sadofyev:2010is (); Sadofyev:2010pr (); Zakharov:2012vv (); Banerjee:2012iz (); Neiman:2010zi (); Jensen:2012jy (). Here we pick up one of these derivations given by Fukushima, Kharzeev, and Warringa Fukushima:2009ft () because it is elementary and easy to see the relation between CME and the lowest Landau level and the axial anomaly.

Let be the thermodynamical potential of fermions of charge in a magnetic field at finite vector and axial chemical potentials, . In the noninteracting limit, can be written as

 Ω=|eB|2π∑s=±∞∑n=0αn,s∫dpz2π{En,s+Tln[1+e−β(En,s−μV)][1+e−β(En,s+μV)]}, (32)

where runs over all the Landau levels and is over spins, is the dispersion relation of the fermions

 En,s=√[sgn(pz)(p2z+2n|eB|)1/2+sμA]2+m2, (33)

and is a degenerate constant given by which accounts the fact that only one spin state occupies the lowest Landau level. The current can be obtained through differentiation of with respect to vector potential , , which owing to gauge invariance is equivalent to in the integrand,

 JzV=e|eB|2π∑s=±∞∑n=0αn,s∫Λ−Λdpz2π∂En,s∂pz[1−nF(En,s+μV)−nF(En,s−μV)], (34)

where is a ultraviolet cutoff that guarantees the finiteness of the calculation at the intermediate steps and goes to infinity at the end of the calculation, and is the Fermi-Dirac distribution. It is then straightforward to find that

 JzV = e|eB|(2π)2∑s=±∞∑n=0αn,s[En,s(pz=Λ)−En,s(pz=−Λ)] (35) = e|eB|(2π)2∑s=±∞∑n=0αn,s[(Λ2+2n|eB|)1/2+sμA−((Λ2+2n|eB|)1/2−sμA)] = e2μA2π2B.

This is just Eq. (30). This derivation shows that: (1) The CME is due to the ultraviolet surface integral and is unaffected by infrared parameters, like , , , etc. (2) Only the lowest Landau level contributes to CME, reflecting the fact that only the lowest Landau level permits a touching node of opposite chirality. This touching node is known to be responsible for the axial anomaly Nielsen:1983rb (); Ambjorn:1983hp (): the applied and fields pump the fermions at LH-chirality branch to RH-chirality branch at the touching node at a rate .

We emphasize again that although the derivation here is for non-interacting system and relies on Landau quantization picture, the CME conductivity is actually fixed by the axial anomaly equation and thus universal no matter how strong the interaction between fermions is (Recall that the axial anomaly equation itself is universal in the sense that it does not receive perturbative correction from scattering between fermions, a result usually referred to as Adler-Bardeen theorem). This is particularly supported by the derivation of CME based on holographic models Newman:2005hd (); Yee:2009vw (); Rebhan:2009vc (); D'Hoker:2009bc (); Rubakov:2010qi (); Gorsky:2010xu (); Gynther:2010ed (); Hoyos:2011us (); Amado:2011zx (); Nair:2011mk (); Kalaydzhyan:2011vx (); Loganayagam:2012pz (); Lin:2013sga () which intrinsically describe strongly coupled system, where although the whole setup is very different from the above derivation and other calculations based on perturbation theory, the CME conductivity is shown to be given by the same universal result.

(3) How can QGP be chiral? — The appearance of CME requires a nonzero which characterizes the strength of the chirality imbalance of the system. Then the question is: how can the QGP generate a net chirality imbalance? To answer this question, following the argument in Ref. Kharzeev:2007jp (), let us first consider the vacuum state of the gauge theory. To make the energy minimized, the vacuum must satisfy the condition ( is the field strength tensor) which requires the gauge field to be pure gauge: with . Working in temporal gauge and by noting that for any time-independent gauge transformation, , the temporal gange fixing condition is unchanged, , one can realize that the vacuum is described by a time-independent which is a pure gauge potential

 Ai(x)=ig−1U−1(x)∂iU(x). (36)

If we impose the boundary condition constant at (see, for example, Refs. Srednicki:2007qs (); Weinberg:1996kr (); Cheng:1985bj (); Nair:2005iw () for relevant discussions about the boundary condition), the gauge transformation defines a map from ( with the infinity identified as an ordinary point) to which is characterized by a winding number ,

 nw=124π2∫d3xϵijktr[(U−1∂iU)(U−1∂jU)(U−1∂kU)]. (37)

This winding number is a topological invariant as can be checked by smoothly deforming . Thus the ’s corresponding to different are topologically distinct in the sense that then cannot be smoothly deformed into each other without passing through gauge field configurations whose field strengths are nonzero. In other words, the ’s of different define multiple degenerate vacua (called the -vacua) separated by finite energy barriers.

On the other hand, one can categorize all the gauge field configurations in to topologically distinct classes characterized by different values of the following topological invariant,

 q=g232π2∫d4xGaμν~Gμνa, (38)

where . This is called the (second) Chern number of configuration and is always an integer Nakahara:2003nw (); Nash:1983cq (). It is straightforward to show that under the condition at infinity ( or ),

 q=124π2∫dΣμϵμνρσtr[(U−1∂νU)(U−1∂ρU)(U−1∂σU)], (39)

where is a surface at infinity in four-dimensional spacetime and satisfies for and constant for  Srednicki:2007qs (); Weinberg:1996kr (); Cheng:1985bj (); Nair:2005iw (). Thus, after several steps of manipulations,

 q=nw(t=∞)−nw(t=−∞). (40)

This means that the gauge field configuration which goes to pure gauge at infinity and has finite can induce a transition from vacuum of winding number to another vacuum of winding number . At zero temperature, such gauge field configurations are called instantons Belavin:1975fg () and they are responsible for the quantum tunneling through the energy barrier between vacua of different winding numbers tHooft:1976up (); tHooft:1976fv (); Jackiw:1976pf (); Callan:1976je ().

The high energy barrier ( MeV) between two vacua suppresses the instanton transition rate exponentially, but at high-enough temperature, the transition between different vacua can also be induced by another, classical, thermal excitation called sphaleron Manton:1983nd (); Klinkhamer:1984di () which, instead of tunneling through the barrier, can take the vacuum over the barrier. In electroweak theory, sphaleron transitions cause baryon number violation and may be important for the cosmological baryogenesis Kuzmin:1985mm (); Rubakov:1996vz (). In QCD, the existence of the sphaleron configurations at finite temperature enormously increases the transition rate McLerran:1990de (); Arnold:1996dy (); Huet:1996sh (); Moore:1997im (); Moore:1997sn (); Bodeker:1998hm (); Bodeker:1999gx (); Shuryak:2001cp (); Ostrovsky:2002cg (); Moore:2010jd (). At very high temperature, the perturbative calculation of the sphaleron transition rate gives  Kharzeev:2007jp (); Arnold:1996dy (); Huet:1996sh (); Moore:1997im (); Moore:1997sn (); Bodeker:1998hm (); Bodeker:1999gx (); Shuryak:2001cp (); Ostrovsky:2002cg (); Moore:2010jd ()

 Γsph∼(αsNc)5T4, (41)

while the strong coupled holographic approach gives an even larger rate  Son:2002sd (). Thus at high temperature, the sphaleron transition rate can be very large. This provides a machinery of generating P and CP odd bubbles in QGP (note that a transition process from a topologically trivial vacuum to a topologically nontrivial vacuum violates P and CP symmetry as is evident from the integrand of ).

Now if we integrate the axial anomaly equation in the QCD sector (for massless quarks)

 ∂μJμA=g2Nf16π2Gaμν~Gμνa, (42)

where ( is over all massless flavors) is the axial current, we see that

 NA(t=∞)−NA(t=−∞)=2q, (43)

where is the total chirality or axial charge. This demonstrates that a topologically nontrivial gauge field configuration can create or annihilate the total chirality of fermions, and thus if the QGP contains a (sufficiently large) domain in which is finite we would expect that it finally will contain unequal numbers of RH and LH quarks or anti-quarks even if initially . This is how QGP can become chiral. We note here that the probabilities of generating positive chirality and negative chirality are equal which means that over many colliding events in heavy-ion collisions the averaged chirality should vanish. What remains after event average is the chirality fluctuation rather the chirality itself and any measurement of the chirality-imbalance effects should be on the event-by-event basis, see Sec. IV.

(4) What is axial chemical potential? — There is a conceptual problem in interpreting as the axial chemical potential for fermions: as when we talk about the chemical potential we always need to associate to it a conserved quantity but we know that the axial charge of fermions is in general not conserved when the fermions are coupled to gauge fields. Furthermore, if is really a chemical potential conjugate to a conserved axial charge of fermions, then Eq. (30) would imply a persistent electric current even at global equilibrium. But the appearance of CME current in equilibrium violates the gauge invariance in the QED sector as discussed in Ref. Kharzeev:2013ffa (), see also Refs. Rebhan:2009vc (); Rubakov:2010qi (); Basar:2013iaa (); Fukushima:2012vr () for relevant discussions. Thus the physical meaning of is actually quite confusing, and there have been a number of relevant discussions in literature Frohlich:2000en (); Vilenkin:1980ft (); Kharzeev:2009fn (); Rebhan:2009vc (); Rubakov:2010qi (); Fukushima:2012vr (); Kharzeev:2013ffa (); Basar:2013iaa (). According to these studies, the CME should be considered as a non-equilibrium phenomenon which vanishes when the system is in global equilibrium and, correspondingly, shouldn’t be regarded as a fixed chemical potential of the fermions in equilibrium. It is more appropriate to view as the rate of the time changing of the field (see below), ; it is the parameter that describes the state of the system.

To reveal the relation between and the field let us consider the -vacua of QCD. The effect of the -vacua can be encoded into a -term in QCD Lagrangian,

 LQCD = −14GaμνGμνa+∑f¯ψf[iγμ(∂μ−igAμ)]ψf−θg2Nf32π2Gaμν~Gμνa, (44)

where is the gluon field. For massless quarks the -term does not give observable consequence because it can be rotated away by transformation, but if we promote the angle to be spacetime dependent (a pseudoscalar ‘axion’ background), then it will show important physical consequences. Perform the path integral over the quarks:

 Z[A,θ]=∫[dψ][d¯ψ]eiS(A). (45)

The -term in the action can be eliminated by performing a local transformation to quark fields and by using the Fujikawa method:

 ψf(x) → eiθ(x)γ5ψf(x), ¯ψf(x) → ¯ψf(x)eiθ(x)γ5. (46)

The resulting Lagrangian reads